# cape: A package for the combined analysis of pleiotropy and epistasis

## Abstract

Here we present an update of the R package for the Combined Analysis of Epistasis and Pleiotropy, or cape. This package implements a method, originally described in Carter et al. (2012), that infers directed interaction networks between genetic variants for predicting the influence of genetic perturbations on quantitative traits. This method takes advantage of complementary information in partially pleiotropic genetic variants to resolve directional influences between variants that interact epistatically. cape can be applied to a variety of genetic variants, such as single nucleotide polymorphisms (SNPs), copy number variations (CNVs) or structural variations (SVs). Here we demonstrate the functionality of cape by inferring a predictive network between quantitative trait loci (QTL) in a cross between the non-obese, non-diabetic (NON) mouse and the New Zealand obese (NZO) mouse (Reifsnyder 2000).

To install cape use:

install.packages("cape")

After installation load cape with the following command:

The current version of cape recognizes data in multiple formats: R/qtl, R/qtl2, and PLINK.

In addition to supporting the newer R/qtl2 (Broman et al. (2019)) format, we also support the R/qtl (Broman et al. (2003)) csv format for cape. You can learn more about this format at the R/qtl website. Briefly, this format contains all traits and genotypes in one comma-delimited file. The first few columns contain traits, covariates, and individual identifiers, and the remaining contain genotypes. Please see the qtl demo data included with cape for an example. Also see ?read_population for more details.

To read in data with this format, we use read_population, followed by cape2mpp, which updates the object to the newer cape format.

The following code is from the demo_qtl example.

results_path <- here::here("demo", "demo_qtl")
data_path <- here::here("tests", "testthat", "testdata", "demo_qtl_data")
data_file <- file.path(data_path, "NON_NZO_Reifsnyder_pgm_CAPE_num.csv")
param_file <- file.path(results_path, "NON_NZO.parameters.yml")

cross <- read_population(data_file)
## The genotypes are encoded as 0, 1, 2.
## Converting to 0, 0.5, 1.
##
## Removing markers on the X chromosome
## Missing values detected in the genotype matrix.
##  If you are planning to use the kinship correction, please use impute.geno() to impute the genotype data.
## Read in the following data:
##  - 208 individuals -
##  - 84 markers -
##  - 8 phenotypes -
cross_obj <- cape2mpp(cross)
obesity_cross <- cross_obj$data_obj obesity_geno <- cross_obj$geno_obj$geno ### Loading data formatted for qtl2 Many researchers are now using the updated R/qtl2 (Broman et al. (2019)), and we also provide ways to handle data in this format. If you are not familiar with qtl2, we highly recommend reading about it and testing it out. R/qtl2 is the definitive genetic mapping package, and the outstanding documentation can answer many basic questions about genetic mapping. The following code shows how to access remote example data provided by Karl Broman for the qtl2 package. For both accessing remote data, or local data, we use the read_cross2 function from qtl2 to read in the data, and then qtl2_to_cape to convert the qtl2 object to a cape object. iron_qtl2 <- read_cross2("https://kbroman.org/qtl2/assets/sampledata/iron/iron.yaml") iron_cape <- qtl2_to_cape(cross = iron_qtl2) data_obj <- iron_cape$data_obj
geno_obj <- iron_cape\$geno_obj 

### Getting started with an example

In this vignette, we will reanalyze a data set described in Reifsnyder (2000). This experiment was performed to identify quantitative trait loci (QTL) for obesity and other risk factors of type II diabetes in a reciprocal back-cross of non-obese non-diabetic NON/Lt mice and the diabetes-prone, New Zealand obese (NZO/HILt) mice. The study found multiple main-effect QTL influencing phenotypes associated with diabetes and obesity as well as multiple epistatic interactions. In addition, maternal environment (i.e. whether the mother was obese) was found to interact with several markers and epistatic pairs to influence the risk of obesity and diabetes of the offspring. The complex nature of diabetes and obesity, along with their complex and polygenic inheritance patterns, make this data set ideal for an analysis of epistasis and pleiotropy.

Included in this dataset are 204 male mice genotyped at 85 MIT markers across the genome. The phenotypes included are the body weight (g), insulin levels (ng/mL), and the log of plasma glucose levels (mg/dL), all measured at age 24 weeks. In addition, there is a variable called pgm’’ indicating whether the mother of each mouse was normal weight (0) or obese (1).

These data can be accessed through the demo_qtl demonstration code.

## Data visualization

Before proceeding with an analysis we recommend exploring the data visually. cape provides a few basic functions for looking at distributions of traits and the correlations between traits.

The figure below shows distributions of three traits: body weight at 24 weeks (BW_24), serum insulin levels (INS_24), and the log of serum glucose levels (log_GLU_24). We have selected these traits out of all traits included in the data set by using the pheno_which argument. Leaving this argument as NULL includes all traits in the plot.

hist_pheno(obesity_cross, pheno_which = c("BW_24", "INS_24", "log_GLU_24"))

Body weight looks relatively normally distributed, but glucose and insulin have obviously non-normal distributions. We can see this in a different way by looking at the Qnorm plots for each trait using qnorm_pheno.’

qnorm_pheno(obesity_cross, pheno_which = c("BW_24", "INS_24", "log_GLU_24"))

In general we recommend mean centering and normalizing all traits before proceeding with the analysis. Trait normalization can be achieved through log transformation, quantile normalization, or another method before the analysis. The function norm_pheno uses quantile normalization to fit the phenotypes to a normal distribution. Briefly, this process ranks the trait values and divides by n-1. It then returns the quantiles of the normal distribution using qnorm. Mean centering subtracts the mean value from each trait, and standardizing divides by the standard deviation.

obesity_cross <- norm_pheno(obesity_cross, mean_center = TRUE)

Now when we plot the distributions compared to a theoretical normal distribution, we see that our traits are much closer to normally distributed. Insulin still has a ceiling effect, which cannot be removed by normalization because rank cannot be determined among equal values.

qnorm_pheno(obesity_cross, pheno_which = c("BW_24", "INS_24", "log_GLU_24"))

## A Note on Trait Selection

Cape relies on the selection of two or more traits that have common genetic factors but are not identical across all individuals. One indirect way at getting at this is through trait correlation. Traits that are modestly correlated may have both common and divergent contributing genetic factors. This is in contrast to traits that are perfectly correlated and likely have only common genetic influences, and to traits that are completely uncorrelated and likely have divergent genetic influences.

We can look at the correlation of the traits in our data set using plot_pheno_cor The figure below shows the correlations between all pairs of traits. Here, we have also colored the points by the covariate “pgm” to look for some initial trends. You can see that mice with obese mothers tended to have higher body weight, insulin levels, and glucose levels.

The lower triangle of panels in the figure below shows the point clouds of each pair of traits plotted against each other. The diagonal shows the distribution of each individual trait, and the upper triangle gives numeric information about pairwise correlations. If color_by is not specified, only the overall pearson R values are shown for each trait pair. If color_by is specified, we show the overall correlation as well as correlations for each group in color_by.

plot_pheno_cor(obesity_cross, pheno_which = c("BW_24", "INS_24", "log_GLU_24"),
color_by = "pgm", group_labels = c("Non-obese", "Obese"))

The traits in this data set are all modestly correlated, which is ideal for cape. In addition, we have selected traits from a range of physiological levels. Body weight is a high-level physiological trait, whereas glucose and insulin levels are endophenotypes measured at the molecular level.

Cape measures interactions between pairs of genes across all traits with the assumption that different genetic interactions identified for a single gene pair in the context of different phenotypes represent multiple manifestations of a single underlying gene network. By measuring the interactions between genetic variants in different contexts we can gain a clearer picture of the network underlying statistical epistasis (Carter et al. 2012).

## Specifying cape parameters

Before starting a cape run, we need to specify all the parameters for the run in a parameter file. We use a .yml file for these parameters. A .yml file holds information easily readable to both humans and computers. We have multiple examples of cape parameter files in the demo folder associated with the cape package. It is probably easiest to start with one of these and modify it to fit your data. The following sections describe each cape parameter in the yml file.

### General parameters

The section of general parameters is where we specify which traits we will use, whether we want cape to normalize the traits for us, and whether we want cape to save the results. This is also the section in which we tell cape whether we want to use a kinship correction or not. kinship corrections are discussed further below.

If a parameter has multiple entries, for example we always want to test multiple traits, each entry gets its own line starting with a dash.

traits: This is where the traits to be scanned are entered. Each trait is entered on its own line preceded by a dash.

covariates: Any covariates are entered here. These are originally part of the trait matrix and are designated by cape as covariates. Each covariate specified is included as an additive covariate in each model, and also tested as an interactive covariate. Here we specify “pgm” as our covariate.

scan_what: This parameter refers to whether we will scan individual traits or composite traits called eigentraits. Eigentraits are calculated by factoring the trait matrix by singular value decomposition (SVD):

$Y = U \cdot V \cdot W^{T}$

Where $$Y$$ is a matrix containing one column for each mean-centered, normalized phenotype and one row for each individual. If $$Y$$ contains more individuals than phenotypes, the $$U$$ matrix has the same dimensions as Y with each column containing one eigentrait. $$V$$ contains the singular values, and $$W^{T}$$ contains the right singular vectors in rows. See Carter et al. (2012) for more details.

The SVD de-correlates the traits concentrating phenotypic features into individual eigentraits. One benefit of this process is that variants that are weakly correlated to several traits due to common underlying processes may be strongly correlated to one of the eigentraits. This eigentrait captures the information of the underlying process, making strong main effects distributed between traits easier to detect and identify as potential interaction loci and/or covariates.

To specify using individual traits, set scan_what to raw_traits. To specify using eigentraits, set scan_what to eigentraits.

traits scaled: This is a logical value (true/false) indicating whether cape should mean center and standardize the traits.

traits normalized: This is a logical value indicating whether cape should normalize the traits.

eig_which: After performing the SVD to create orthogonal eigentraits, you may wish to analyze only a subset of them in cape. For example, if the first two eigentraits capture more than 90% of the trait variance, you may wish to discard the last eigentrait. This results in a loss of some information, but may increase power to detect important trait-related signals.

To specify which eigentraits to use, enter each on its own line. This is a bit more tedious than simply entering a number of eigentraits, but offers a little more flexibility in case you would like to analyze eigentraits that don’t necessarily start at eigentrait 1.

pval_correction: This is where we specify the correction for multiple testing. It can be any of the options in p.adjust: “holm”, “hochberg”, “hommel”, “bonferroni”, “BH”, “BY”, “fdr”, or “none.”

use_kinship: A logical value indicating whether to use a kinship correction.

pop: This parameter is only necessary to set if use_kinship is set to true. If use_kinship is true, pop can be one of “2PP” for a two-parent cross, “MPP” for a multi-parent cross, or “RIL” for a recombinant inbred line. This parameter is passed to kinship to inform the calculation of the kinship matrix.

save_results: A logical value indicating whether results should be written to files.

use_saved_results A logical value indicating whether cape should use data already written to file. This can be helpful if a job aborts partway through. You can use everything that has been calculated already and save time on the next run. If use_saved_results is true, only results that don’t already exist will be calculated. This can cause problems if there has been a parameter change between two runs, and there is a mismatch between previously calculated results and new results. We recommend setting this to false for most cases.

transform_to_phenospace A logical value indicating that is required scanning eigentraits. If true, all effects are rotate back to trait space after calculation, such that the final networks show marker influences on traits. If false, all effects will be kept in terms of eigentraits, and all final tables and plots will show marker effects on eigentraits.

### Single scan parameters

Cape performs marker regression on individual markers before running the pairwise tests. If there are more markers than can be tested reasonably in the pairwise scan, cape uses the results from the single-locus scan to select markers for the pairwise scan.

ref_allele: The reference allele. In two-parent populations, this will usually be allele A (major allele). In multi-parent populations, this may be a different allele. For example, the C57B6/J mouse is often used as a reference strain in mouse studies. It is also one of the founders of the Diversity Outbred (DO) and Collaborative Cross (CC) mice (Chesler et al. 2008), and serves as a reasonable reference. To specify the B6 mouse as the reference, use its letter code “B” as the reference allele. All alleles in cape are stored as individual letters, A through the number of alleles that are present. For example, there are eight founder alleles in the DO/CC mice that are represented by the letters A through H.

singlescan_perm: This parameter specifies the number of permutations done in the single-locus scan. It is perfectly okay to set this to 0. The permutations are not used for anything in cape. They simply allow you to make a first-pass glance at identifying markers with main effects in your data. However, if you are interested in doing the main effect scan properly, we recommend using R/qtl2 (Broman et al. 2019).

alpha: If “singlescan_perm” is a number greater than 0, you can also specify a significance threshold alpha. If alpha is specified, cape will calculate an effect size threshold for each value of alpha. These can be plotted using plot_singlescan.

### Marker Selection Parameters

As mentioned above, the single-locus scan is primarily used to select markers for the pairwise scan. We usually do this by selecting the markers with the largest main effects. Alternatively, you can also specify markers for the pairwise scan using a text file.

marker_selection_method: This parameter should be either “top_effects”, or “from_list.” If “top_effects,” cape will select markers for the pair scan based on their main effects across all traits in the single-locus scan. In this case, a few additional parameters need to be specified:

peak_density: To select markers for the pairscan, cape first identifies peaks in effect size across all traits, and samples markers within the peaks for further testing. “peak_density” specifies the fraction of markers that will be sampled uniformly from under a large peak. For example, if “peak_density” is set to 0.5, 50% of markers under a peak will be selected for downstream testing. For populations with high linkage disequilibrium (LD), you might want to set this lower to get a better sampling of peaks from across the genome with less redundancy. For a population with low LD, you may want to set this parameter higher to get a better sampling of markers with large main effects.

num_alleles_in_pairscan: This parameter sets how many markers will be tested in the pairwise scan. Cape is relatively computationally intensive, and we cannot do exhaustive pairwise testing in densely genotyped populations. We suggest limiting this number to around 1500 markers. These can be tested in roughly 24 hours using 2 or 3 traits/eigentraits and 1.5 million permuted tests. Increasing the number of traits/eigentraits analyzed substantially increases the time of a run, and it might be necessary to lower the number of markers tested further if there are many traits/eigentraits in the analysis.

tolerance: In selecting markers by their main effect size, cape starts at a high effect size, and gradually lowers it until it has collected the desired number of markers. The “tolerance” gives cape some wiggle room in how many markers are selected. If “num_alleles_in_pairscan” is 100 and the “tolerance” is 5, cape will allow any number of markers between 95 and 105.

snp_file: If “marker_selection_method” is “from_list” a file name needs to be specified. The file must be in the results directory. It is a tab-delimited text file with one column. The column should contain the names of the markers to be tested in the pairscan. This can get a little tricky for multi-parent crosses. Because cape tests individual alleles and not markers in the pair scan, the markers must also have the allele name appended with an underscore ("_“). This is relatively simple for a two-parent cross: If the reference allele is A, all markers will just have a”_B" appended to their name. If, however, this is a multi-parent population, specifying the correct allele is important. Any of the alleles other than the reference can be specified for each marker. The underscore introduces another little complication, which is that no underscores are allowed in marker names except to separate the marker name from the allele. If there are underscores in marker names, cape deletes these. This means that any file specifying marker names for the pairscan should also remove all underscores that are not separating the marker name from the allele. This is a bit cumbersome, but can be completely avoided by setting “marker_selection_method” to “top_effects.”

### Pairscan Parameters

The pairwise scan has a few additional parameters to set. Because testing genetic markers in LD can lead to false positives, we refrain from testing marker pairs that are highly correlated. This (Pearson) correlation is set by the user as “max_pair_cor”. We generally use a value of 0.5 in our own work, but this can be raised or lowered depending on your own assessment of the influence of LD in your population.

max_pair_cor: A numeric value between 0 and 1 indicating the maximum Pearson correlation that two markers are allowed in pairwise testing. If the correlation between a pair of markers exceeds this threshold, the pair is not tested. If this value is set to NULL, “min_per_genotype” must have a numeric value.

min_per_geno: This is an alternative way to limit the effect of LD on pairwise testing. This number sets the minimum number of individuals allowable per pairwise genotype combination. If for a given marker pair, one of the genotype combinations is underrepresented, the marker pair is not tested. Setting this parameter is not recommended if you have continuous genotype probabilities, as the number of genotype combinations will be too high. If this value is NULL, max_pair_cor must have a numeric value.

pairscan_null_size: This parameter can be a little confusing, as it is different from the number of permutations to run in the the pairwise scan. Carter et al. (2012) showed that multiple tests from a single permutation can be combined to create the null distribution for cape statistics. This saves enormously on time. So instead of specifying the number of permutations, we specify the total size of the null distribution. If you are testing 100 markers, you will test at most 100 choose 2, or 4950 marker pairs. Choose a “pairscan_null_size” that is at least as big as the number of marker pairs you are testing. For smaller numbers of markers, you may want to choose a null distribution size that is many times larger than the number of pairs you are testing.

## Running cape

Unlike previous versions of cape, which required the user to perform many individual steps, this version of cape can be run with a single command run_cape. Now that we’ve done the hard work of reading in our data and specifying our parameters, we can sit back and let the computer take care of the rest. The line below runs the complete cape pipeline for the demo NON/NZO mouse backcross. If you run this line interactively, set verbose to TRUE to see the progress printed to your screen. When verbose is set to FALSE, important messages are still printed, but printing of progress is suppressed.

final_cross <- run_cape(obesity_cross, obesity_geno, results_file = "NON_NZO.RData",
p_or_q = 0.05, verbose = FALSE, param_file = param_file, results_path = results_path)
## Removing 18 individuals with missing phenotypes.

The benefit to this new function is obvious: cape is now much easier to run. However, it comes with a sizeable reduction in flexibility. For most users the benefit of the increased ease of use will far outweigh the decrease in flexibility.

If you do need of more flexibility, all functions in the function run_cape are exported, so the entire analysis can be run step-by-step as it was in earlier versions. All plotting functions are also exported to allow greater flexibility in plotting. Although run_cape does some plotting automatically, you may wish to re-run some plotting functions with your own settings.

However you choose to run cape, the cape team is very happy to answer any questions and to help with running or interpreting any cape analysis.

### Eigentraits

As described above, the first step performed by cape is typically to decompose the trait matrix into eigentraits. This is done if “scan_what” is set to “eigentraits.”

This step uses singular value decomposition (SVD) to decompose the trait matrix into orthogonal eigentraits. Because we use modestly correlated traits in cape by design, the eigentrait decomposition may help concentrate correlated signals, that are otherwise distributed across traits, into individual eigentraits. This potentially increases power to detect variants associated with the common components of groups of traits.

Cape uses the function plot_svd to plot the results of the SVD. The plot for the NON/NZO data set are shown below.

plot_svd(final_cross)

In the example illustrated here, the first eigentrait captures 75% of the total trait variance. This eigentrait describes the processes by which body weight, glucose levels, and insulin levels all vary together. The correlations between obesity and risk factors for obesity, such as elevated insulin and fasting glucose levels are well known (Permutt, Wasson, and Cox 2005; Das and Elbein 2006; Haffner 2003). The second eigentrait captures 14% of the variance in the phenotypes. It captures the processes through which glucose and body weight vary in opposite directions. This eigentrait may be important in distinguishing the genetic discordance between obesity and diabetes. While obesity is a strong risk factor for diabetes, not all those who are obese have diabetes, and not all those with diabetes are obese (Permutt, Wasson, and Cox 2005; Burcelin et al. 2002).

The third eigentrait is less interpretable biologically, as it describes the divergence of blood glucose and insulin levels. It may represent a genetic link between glucose and body weight that is non-insulin dependent. Because we are primarily interested in the connection between diabetes and insulin, we used only the first two eigentraits for the analysis. In many cases in which more than two phenotypes are being analyzed, the first two or three eigentraits will capture the majority of the variance in the data and capture obvious features. Other eigentraits may capture noise or systematic bias in the data. Often the amount of total variance captured by such eigentraits is small, and they can be removed from the analysis.

Ultimately, there is no universal recipe for selecting which eigentraits should be included in the analysis, and the decision will be based on how the eigentraits contribute to the original phenotypes and how much variance in the data they capture.

### Single-locus scan

After computing eigentraits, we go on to perform marker regression on all individual markers:

$U_{i}^{j} = \beta_{0}^{j} + x_{i}\beta^{j} + \epsilon_{i}^{j}$

The index $$i$$ runs from 1 to the number of individuals, and $$j$$ runs from 1 to the number of eigentraits or traits. $$x_{i}$$ is the probability of the presence of the reference allele for individual $$i$$ at locus $$j$$. The primary purpose of this scan is is to select markers for the pairwise scan if there are too many to test exhaustively.

Although run_cape creates files with the singlescan results automatically, it is also sometimes desireable to re-plot these results with different parameters. We show how to do that below by reading in the singlescan results file and using plot_singlescan with parameters different from those in run_cape.

singlescan_obj <- readRDS(here::here("demo", "demo_qtl", "NON_NZO_singlescan.RData"))
plot_singlescan(final_cross, singlescan_obj, line_type = "h", lwd = 2,
covar_label_size = 1)`